Testing polynomials over general fields

In this work we fill the knowledge gap concerning testing polynomials over finite fields. As previous works show, when the cardinality of the field, q, is sufficiently larger than the degree bound, d, then the number of queries sufficient for testing is polynomial or even linear in d. On the other hand, when q = 2 then the number of queries, both sufficient and necessary, grows exponentially with d. Here we study the intermediate case where 2 < q = O(d) and show a smooth transition between the two extremes. Specifically, let p be the characteristic of the field (so that p is prime and q = p^s for some integer s ≥ 1). Then the number of queries performed by the test grows like ℓ · q^2ℓ+1, where ℓ = [d+1/q-q/p]. Furthermore, 9^Ω(ℓ) queries are necessary when q = O(d). The test itself provides a unifying view of the tests for these two extremes: it considers random affine subspaces of dimension ℓ and verifies that the function restricted to the selected subspaces is a polynomial of degree at most d. Viewed in the context of coding theory, our result shows that Reed-Muller codes over general fields (usually referred to as generalized Reed-Muller (GRM) codes) are locally testable. In the course of our analysis we provide a characterization of small-weight words that span the code. Such a characterization was previously known only when the field size is a prime or is sufficiently large, in which case the minimum-weight words span the code.